Random Matrix Results Research Articles

Stability-instability transition in tripartite merged ecological networks

Although ecological networks are typically constructed based on a single type of interaction, e.g. trophic interactions in a food web, a more complete picture of ecosystem composition and functioning arises from merging networks of multiple interaction types. In this work, we consider tripartite networks constructed by merging two bipartite networks, one mutualistic and one antagonistic. Taking the interactions within each sub-network to be distributed randomly, we consider the stability of the dynamics of the network based on the spectrum of its community matrix. In the asymptotic limit of a large number of species, we show that the spectrum undergoes an eigenvalue phase transition, which leads to an abrupt destabilisation of the network as the ratio of mutualists to antagonists is increased. We also derive results that show how this transition is manifest in networks of finite size, as well as when disorder is introduced in the segregation of the two interaction types. Our random-matrix results will serve as a baseline for understanding the behaviour of merged networks with more realistic structures and/or more detailed dynamics.

Three-fold way of entanglement dynamics in monitored quantum circuits

We investigate the measurement-induced entanglement transition in quantum circuits built upon Dyson’s three circular ensembles (circular unitary, orthogonal, and symplectic ensembles; CUE, COE and CSE). We utilise the established model of a one-dimensional circuit evolving under alternating local random unitary gates and projective measurements performed with tunable rate, which for gates drawn from the CUE is known to display a transition from extensive to intensive entanglement scaling as the measurement rate is increased. By contrasting this case to the COE and CSE, we obtain insights into the interplay between the local entanglement generation by the gates and the entanglement reduction by the measurements. For this, we combine exact analytical random-matrix results for the entanglement generated by the individual gates in the different ensembles, and numerical results for the complete quantum circuit. These considerations include an efficient rephrasing of the statistical entangling power in terms of a characteristic entanglement matrix capturing the essence of Cartan’s KAK decomposition, and a general result for the eigenvalue statistics of antisymmetric matrices associated with the CSE.

Hydrodynamic Theory of the Connected Spectral form Factor

One manifestation of quantum chaos is a random-matrix-like fine-grained energy spectrum. Prior to the inverse level spacing time, random matrix theory predicts a “ramp” of increasing variance in the connected part of the spectral form factor. However, in realistic quantum chaotic systems, the finite-time dynamics of the spectral form factor is much richer, with the pure random matrix ramp appearing only at sufficiently late time. In this article, we present a hydrodynamic theory of the connected spectral form factor prior to the inverse level spacing time. We first derive a general formula for the spectral form factor of a system with almost-conserved sectors in terms of return probabilities and spectral form factors within each sector. Next we argue that the theory of fluctuating hydrodynamics can be adapted from the usual Schwinger-Keldysh contour to the periodic time setting needed for the spectral form factor, and we show explicitly that the general formula is recovered in the case of energy diffusion. We also initiate a study of interaction effects in this modified hydrodynamic framework and show how the Thouless time defined as the time required for the spectral form factor to approach the pure random matrix result is controlled by the slow hydrodynamics modes. We then extend the formalism to Floquet systems, where a ramp is expected but with a different coefficient, and we derive a crossover formula from the Hamiltonian ramp to the Floquet ramp when the Floquet drive is weak. Taken together, these results establish an effective field theory of chaotic spectral correlations which predicts the random matrix ramp at late time and computes corrections to it at earlier times.6 MoreReceived 15 December 2020Revised 12 January 2022Accepted 10 February 2022DOI:https://doi.org/10.1103/PhysRevX.12.021009Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.Published by the American Physical SocietyPhysics Subject Headings (PhySH)Research AreasQuantum chaosPhysical Systems1-dimensional spin chainsTechniquesRandom matrix theorySachdev-Ye-Kitaev modelU(N) symmetryCondensed Matter, Materials & Applied PhysicsNonlinear DynamicsFluid Dynamics

Symplectic n-level densities with restricted support

In this paper, we demonstrate that the alternative form, derived by us in an earlier paper, of the [Formula: see text]-level densities for eigenvalues of matrices from the classical compact group [Formula: see text] is far better suited for comparison with derivations of the [Formula: see text]-level densities of zeros in the family of Dirichlet [Formula: see text]-functions associated with real quadratic characters than the traditional determinantal random matrix formula. Previous authors have found ingenious proofs that the leading order term of the [Formula: see text]-level density of the zeros agrees with the determinantal random matrix result under certain conditions, but here we show that comparison is more straightforward if the more suitable form of the random matrix result is used. For the support of the test function in [Formula: see text] and in [Formula: see text] we compare with existing number theoretical results. For support in [Formula: see text] no rigorous number theoretical result is known for the [Formula: see text]-level densities, but we derive the densities here using random matrix theory in the hope that this may make the path to a rigorous number theoretical result clearer.

Linear precoding based on polynomial expansion: reducing complexity in massive MIMO.

Massive multiple-input multiple-output (MIMO) techniques have the potential to bring tremendous improvements in spectral efficiency to future communication systems. Counterintuitively, the practical issues of having uncertain channel knowledge, high propagation losses, and implementing optimal non-linear precoding are solved more or less automatically by enlarging system dimensions. However, the computational precoding complexity grows with the system dimensions. For example, the close-to-optimal and relatively “antenna-efficient” regularized zero-forcing (RZF) precoding is very complicated to implement in practice, since it requires fast inversions of large matrices in every coherence period. Motivated by the high performance of RZF, we propose to replace the matrix inversion and multiplication by a truncated polynomial expansion (TPE), thereby obtaining the new TPE precoding scheme which is more suitable for real-time hardware implementation and significantly reduces the delay to the first transmitted symbol. The degree of the matrix polynomial can be adapted to the available hardware resources and enables smooth transition between simple maximum ratio transmission and more advanced RZF.By deriving new random matrix results, we obtain a deterministic expression for the asymptotic signal-to-interference-and-noise ratio (SINR) achieved by TPE precoding in massive MIMO systems. Furthermore, we provide a closed-form expression for the polynomial coefficients that maximizes this SINR. To maintain a fixed per-user rate loss as compared to RZF, the polynomial degree does not need to scale with the system, but it should be increased with the quality of the channel knowledge and the signal-to-noise ratio.

Efficient semiclassical approach for time delays

The Wigner time delay, defined by the energy derivative of the total scattering phase shift, is an important spectral measure of an open quantum system characterizing the duration of the scattering event. It is proportional to the trace of the Wigner–Smith matrix Q that also encodes other time-delay characteristics. For chaotic cavities, these quantities exhibit universal fluctuations that are commonly described within random matrix theory. Here, we develop a new semiclassical approach to the time-delay matrix which is formulated in terms of the classical trajectories that connect the exterior and interior regions of the system. This approach is superior to previous treatments because it avoids the energy derivative. We demonstrate the methodʼs efficiency by going beyond previous work in establishing the universality of time-delay statistics for chaotic cavities with perfectly connected leads. In particular, the moment generating function of the proper time-delays (eigenvalues of Q) is found semiclassically for the first five orders in the inverse number of scattering channels for systems with and without time-reversal symmetry. We also show the equivalence of random matrix and semiclassical results for the second moments and for the variance of the Wigner time delay at any channel number.

Combinatorial theory of the semiclassical evaluation of transport moments II: Algorithmic approach for moment generating functions

Electronic transport through chaotic quantum dots exhibits universal behaviour which can be understood through the semiclassical approximation. Within the approximation, calculation of transport moments reduces to codifying classical correlations between scattering trajectories. These can be represented as ribbon graphs and we develop an algorithmic combinatorial method to generate all such graphs with a given genus. This provides an expansion of the linear transport moments for systems both with and without time reversal symmetry. The computational implementation is then able to progress several orders further than previous semiclassical formulae as well as those derived from an asymptotic expansion of random matrix results. The patterns observed also suggest a general form for the higher orders.

Multi-Cell MIMO Downlink With Cell Cooperation and Fair Scheduling: A Large-System Limit Analysis

We consider the downlink of a cellular network with multiple cells and multi-antenna base stations, including a realistic distance-dependent pathloss model, clusters of cooperating cells, and general "fairness" requirements. Beyond Monte Carlo simulation, no efficient computation method to evaluate the ergodic throughput of such systems has been presented so far. We propose an analytic solution based on the combination of large random matrix results and convex optimization. The proposed method is computationally much more efficient than Monte Carlo simulation and provides surprisingly accurate approximations for the actual finite-dimensional systems, even for a small number of users and base station antennas. Numerical examples include 2-cell linear and three-sectored 7-cell planar layouts, with no inter-cell cooperation, sector cooperation, or full inter-cell cooperation.

Moments of the Wigner delay times

The Wigner time delay is a measure of the time spent by a particle inside the scattering region of an open system. For chaotic systems, the statistics of the individual delay times (whose average is the Wigner time delay) are thought to be well described by random matrix theory. Here we present a semiclassical derivation showing the validity of random matrix results. In order to simplify the semiclassical treatment, we express the moments of the delay times in terms of correlation functions of scattering matrices at different energies. In the semiclassical approximation, the elements of the scattering matrix are given in terms of the classical scattering trajectories, requiring one to study correlations between sets of such trajectories. We describe the structure of correlated sets of trajectories and formulate the rules for their evaluation to the leading order in inverse channel number. This allows us to derive a polynomial equation satisfied by the generating function of the moments. Along with showing the agreement of our semiclassical results with the moments predicted by random matrix theory, we infer that the scattering matrix is unitary to all orders in the semiclassical approximation.

Concentration of measure and spectra of random matrices: Applications to correlation matrices, elliptical distributions and beyond

We place ourselves in the setting of high-dimensional statistical inference, where the number of variables p in a data set of interest is of the same order of magnitude as the number of observations n. More formally, we study the asymptotic properties of correlation and covariance matrices, in the setting where p/n→ρ∈(0, ∞), for general population covariance. We show that, for a large class of models studied in random matrix theory, spectral properties of large-dimensional correlation matrices are similar to those of large-dimensional covarance matrices. We also derive a Marcenko–Pastur-type system of equations for the limiting spectral distribution of covariance matrices computed from data with elliptical distributions and generalizations of this family. The motivation for this study comes partly from the possible relevance of such distributional assumptions to problems in econometrics and portfolio optimization, as well as robustness questions for certain classical random matrix results. A mathematical theme of the paper is the important use we make of concentration inequalities.

Semiclassical approach to the ac conductance of chaotic cavities

We address frequency-dependent quantum transport through mesoscopic conductors in the semiclassical limit. By generalizing the trajectory-based semiclassical theory of dc quantum transport to the ac case, we derive the average screened conductance as well as ac weak-localization corrections for chaotic conductors. Thereby we confirm respective random matrix results and generalize them by accounting for Ehrenfest time effects. We consider the case of a cavity connected through many leads to a macroscopic circuit which contains ac-sources. In addition to the reservoir the cavity itself is capacitively coupled to a gate. By incorporating tunnel barriers between cavity and leads we obtain results for arbitrary tunnel rates. Finally, based on our findings we investigate the effect of dephasing on the charge relaxation resistance of a mesoscopic capacitor in the linear low-frequency regime.

Design of cyclic delay diversity for single carrier cyclic prefix (SCCP) transmissions with block-iterative GDFE (BI-GDFE) receiver

<i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Single</i> <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">carrier</i> <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">cyclic</i> <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">prefix</i> (SCCP) modulation has been adopted as one of the physical layer air-interfaces for IEEE802.I6 standards due to its less stringent requirement on <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">peak-to-average</i> <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">power</i> <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ratio</i> (PAPR) than multicarrier modulation. As a single carrier transmission, however, the received SCCP blocks suffer from inter-symbol interference (ISI). The <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">block</i> <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">iterative</i> <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">generalized</i> <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">decision</i> <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">feedback</i> <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">equalizer</i> (BI-GDFE) is an effective interference cancellation scheme which generally shows performance improvement over linear equalizers. However, in order to achieve a significant gain, the channel must contain rich multipath components, and this may not necessarily be true in many systems. In this paper, we propose to apply <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">cyclic</i> <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">delay</i> <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">diversity</i> (CDD) to a SCCP system so that the possible performance improvement by using a BI-GDFE detector can be enhanced. The asymptotic performance of the proposed system is analyzed using random matrix results. We propose a scheme to determine the optimal delay parameter for CDD. Computer simulations have shown that by incorporating CDD into the transmitter and using a BI-GDFE receiver, the BER performance of the system can approach the MFB within several iterations for high SNR region.

Long-time signatures of short-time dynamics in decaying quantum-chaotic systems

We analyze the decay of classically chaotic quantum systems in the presence of fast ballistic escape routes on the Ehrenfest time scale. For a continuous excitation process, the form factor of the decay cross section deviates from the universal random-matrix result on the Heisenberg time scale, i.e., for times much larger than the time for ballistic escape. We derive an exact analytical description and compare our results with numerical simulations for a dynamical model.

A new application of random matrices: Ext(Cred∗(F2)) is not a group

In the process of developing the theory of free probability and free entropy, Voiculescu introduced in 1991 a random matrix model for a free semicircular system. Since then, random matrices have played a key role in von Neumann algebra theory (cf. [V8], [V9]). The main result of this paper is the following extension of Voiculescui¯s random matrix result: Let (X(n) 1 , . . . , X(n) r ) be a system of r stochastically independent n i? n Gaussian self-adjoint random matrices as in Voiculescui¯s random matrix paper [V4], and let (x1, . . . , xr) be a semi-circular system in a C.-probability space. Then for every polynomial p in r noncommuting variables lim niaeiA

pX(n) 1 (¥o), . . . , X(n) r (¥o)

= p(x1, . . . , xr), for almost all ¥o in the underlying probability space. We use the result to show that the Ext-invariant for the reduced C.-algebra of the free group on 2 generators is not a group but only a semi-group. This problem has been open since Anderson in 1978 found the first example of a C.-algebra A for which Ext(A) is not a group.

Design of Reduced-Rank MMSE Multiuser Detectors Using Random Matrix Methods

Reduced-rank minimum mean-squared error (MMSE) multiuser detectors using asymptotic weights have been shown to reduce receiver complexity while maintaining good performance in long-sequence code-division multiple-access (CDMA) systems. In this paper, we consider the design of reduced-rank MMSE receivers in a general framework which includes fading, single and multiantenna receivers, as well as direct-sequence CDMA (DS-CDMA) and multicarrier CDMA (both uplink and downlink). In all these cases, random matrix results are used to obtain explicit expressions for the asymptotic eigenvalue moments of the interference autocorrelation matrix and for the asymptotic weights used in the reduced-rank receiver.

Long-range interactions in the quantum many-body problem in one dimension: ground state.

We investigate the ground state properties of a family of N-body systems in one dimension, trapped in a polynomial potential and having long-range two-body interaction in addition to the inverse square potential studied in the Calogero-Sutherland model (CSM). We show that for such a Hamiltonian, the ground state energy is similar to that of free fermions in a harmonic well with a displacement that depends on the number of particles and depth of the well. We obtain the ground state wave function and using random matrix results, study the particle density and pair correlation function (PCF). We observe that the particles are arranged in bands. Due to the presence of long-range interaction, the PCF shows a departure from the CSM.

Distribution of reflection coefficients in absorbing chaotic microwave cavities.

The distribution of reflection coefficients P(R) for chaotic microwave cavities with time-reversal symmetry is investigated in different absorption and antenna coupling regimes. For all regimes the agreement between experimental distributions and random-matrix theory predictions is very good, provided both the antenna coupling T(a) and the wall absorption strength T(w) are taken into account in an appropriate way. These parameters are determined by independent experimental quantities.

Value Distribution of the Eigenfunctions and Spectral Determinants of Quantum Star Graphs

We compute the value distributions of the eigenfunctions and spectral determinant of the Schrodinger operator on families of star graphs. The values of the spectral determinant are shown to have a Cauchy distribution with respect both to averages over bond lengths in the limit as the wavenumber tends to infinity and to averages over wavenumber when the bond lengths are fixed and not rationally related. This is in contrast to the spectral determinants of random matrices, for which the logarithm is known to satisfy a Gaussian limit distribution. The value distribution of the eigenfunctions also differs from the corresponding random matrix result. We argue that the value distributions of the spectral determinant and of the eigenfunctions should coincide with those of Seba-type billiards.

Noiseless scattering states in a chaotic cavity

Shot noise in a chaotic cavity (Lyapunov exponent $\lambda$, level spacing $\delta$, linear dimension $L$), coupled by two $N$-mode point contacts to electron reservoirs, is studied as a measure of the crossover from stochastic quantum transport to deterministic classical transport. The transition proceeds through the formation of {\em fully} transmitted or reflected scattering states, which we construct explicitly. The fully transmitted states contribute to the mean current $\bar{I}$, but not to the shot-noise power $S$. We find that these noiseless transmission channels do not exist for $N\alt\sqrt{k_{F}L}$, where we expect the random-matrix result $S/2e\bar{I}=1/4$. For $N\agt\sqrt{k_{F}L}$ we predict a suppression of the noise $\propto (k_{F}L/N^{2})^{N\delta/\pi\hbar\lambda}$. This nonlinear contact dependence of the noise could help to distinguish ballistic chaotic scattering from random impurity scattering in quantum transport.

Energy level statistics in weakly disordered systems: from quantum to diffusive regime

We calculate two-point energy level correlation function in weakly disorderd metallic grain with taking account of localization corrections to the universal random matrix result. Using supersymmetric nonlinear sigma model and exactly integrating out spatially homogeneous modes, we derive the expression valid for arbitrary energy differences from quantum to diffusive regime for the system with broken time reversal symmetry. Our result coincides with the one obtained by Andreev and Altshuler [Phys. Rev. Lett. 72, 902 (1995)] where homogeneous modes are perturbatively treated.